Doctoral Thesis in Chemical Engineering
State Estimation of Lithium-ion
Batteries
XIAOLEI BIAN
kth royal institute of technology
State Estimation of Lithium-ion
Batteries
XIAOLEI BIAN
Doctoral Thesis in Chemical Engineering KTH Royal Institute of Technology Stockholm, Sweden 2021
Academic Dissertation which, with due permission of the KTH Royal Institute of Technology, is submitted for public defence for the Degree of Doctor of Philosophy on Wednesday the 9th June 2021, at 2:00 p.m. in Patos, Teknikringen 42, Stockholm.
© Xiaolei Bian ISBN 978-91-7873-840-3 TRITA-CBH-FOU-2021:19
To my wife
Hongdi
Abstract
To guarantee the safety operation, the key states of lithium-ion battery, e.g., the state of charge and the state of health, must be estimated and monitored accurately. This thesis is mainly to develop models and algorithms to accurately and robustly estimate the key battery states, based on the available measurements i.e., current and voltage. All the work is based on four published papers and can be divided into three parts.
The first part of this work presents a two-step parameter optimization method for online state of charge estimation of lithium-ion battery. The particle swarm optimization is exploited for model parametrization and extended Kalman filter tuning. Within this particle swarm optimization-based framework, the searching boundary is derived by scrutinizing the error transition property of the test system, which can narrow the searching region and increase the computational efficiency. In general, the proposed method can well exploit the potential of model-based estimators, leading to a robust model compatibility and optimized performance.
In the second part of this thesis, two novel models are developed to estimate the state of health of lithium-ion battery. The first one is an open circuit voltage-based model, which describes the open circuit voltage as a function of the state of charge by a polynomial, with a lumped thermal model to account for the effect of temperature. It requires a prior learning from the initial constant-current profile. The second model is an incremental capacity analysis-based model, which defines the dependence of the state of charge on the open circuit voltage using a capacity model. It can be learning-free, with the parameters subject to certain constraints. Both models use an equivalent circuit model to characterize the constant-current profiles and a nonlinear least squares method to identify the involving parameters. These two models are validated by aging experiments, and the results show that both can give accurate state-of-health estimation.
The third part of the thesis introduces a fusion-type state-of-health estimator by combining the model-based profile reconstruction and the incremental capacity analysis-based state estimation. The above-mentioned open circuit voltage-based model is employed here to mitigate the noise-induced unfavorable numerical conditions and to modify the incremental capacity curves. Leveraging the modified incremental capacity curves, a set of feature-of-interests are extracted and evaluated, and several cautiously selected ones are used to estimate the state of health of lithium-ion battery. Long-term cycling tests on different lithium-ion batteries are used for validation. This fusion-type method has comparable accuracy and better robustness, compared with the model-based methods. Moreover, the proposed estimator has a good generality to different batteries and also promises an excellent robustness against cell inconsistency, noise corruption, temperature variety, and profile partialness.
Keywords: Lithium-ion battery, State of charge, State of health, Model-based method, Extended Kalman filter, Filter tuning, Fusion model.
Sammanfattning
För att kunna garantera säker användning måste viktiga tillstånd för litiumjonbatterier, t.ex. laddningstillstånd och hälsotillstånd, uppskattas och noggrant övervakas. Syftet med denna avhandling är främst att utveckla modeller och algoritmer för att med hög noggrannhet och robust kunna uppskatta de viktigaste batteristatusarna, med hjälp av tillgängliga mätstorheter, dvs ström och spänning. Arbetet bygger på fyra publicerade artiklar och kan delas in i tre delar. Den första delen av arbetet presenterar en metod för optimering av parametrar i två steg för uppskattning av litiumjonbatteriers laddningstillstånd online. Partikelsvärmsoptimering utnyttjas för modellparametrisering och förlängd Kalmanfilterjustering. Inom detta
partikel-svärmoptimeringsbaserade ramverk härleds sökgränsen genom att granska
felsövergångsegenskapen hos testsystemet, vilket kan begränsa sökningsområdet och öka beräkningseffektiviteten. Generellt kan den föreslagna metoden utnyttja potentialen hos modellbaserade skattningar väl, vilket leder till en robust modellkompatibilitet och optimerad prestanda.
I den andra delen av denna avhandling har två nya modeller för att uppskatta litiumjonbatteriers hälsotillstånd utvecklats. Den första är en spänningsbaserad modell med öppen krets, som beskriver den öppna kretsspänningen som en funktion av laddningstillståndet med ett polynom, med en sammansatt termisk modell för att redogöra för effekten av temperatur. Det kräver en tidigare inlärning från den ursprungliga konstantströmsprofilen. Den andra modellen är en inkrementell kapacitetsanalysbaserad modell, som definierar laddningstillståndets beroende av den öppna kretsspänningen med hjälp av en kapacitetsmodell. Den kan vara inlärningsfri med parametrar som är föremål för vissa begränsningar. Båda modellerna använder en motsvarande kretsmodell för att karakterisera konstantströmsprofilerna och en icke-linjär minsta kvadratmetod för att identifiera parametrar. Dessa två modeller valideras av åldringsexperiment, och resultaten visar att de båda kan ge en god uppskattning av hälsotillstånd.
Den tredje delen av avhandlingen introducerar en fusionstypsbaserad uppskattningsmetod av hälsotillstånd genom att kombinera den modellbaserade profilrekonstruktionen och den
inkrementella kapacitetsanalysbaserade tillståndsuppskattningen. Den ovannämnda
spänningsbaserade modellen med öppen krets används här för att mildra de brusinducerade ogynnsamma numeriska förhållandena och för att modifiera de inkrementella kapacitetskurvorna. Genom att använda de modifierade inkrementella kapacitetskurvorna extraheras och utvärderas en uppsättning funktioner av intresse och flera noga utvalda används för att uppskatta litiumjonbatteriets hälsotillstånd. Cykliska tester, utförda under lång tid, på olika litiumjonbatterier används för validering. Denna fusionsmetod har jämförbar noggrannhet samt bättre robusthet jämfört med modellbaserade metoder. Dessutom är den föreslagna uppskattningsmetoden generell och kan användas för olika batterier. Den visar även på en utmärkt robusthet mot cellinkonsekvens, bruskorruption, temperaturvariation och profilpartialitet.
Nyckelord: Litiumjonbatteri, Laddningstillstånd, Hälsotillstånd, Modellbaserad metod, Utökat Kalman-filter, Filterinställning, Fusionsmodell.
List of Publications
Publications included in this thesis
Paper 1. X. Bian, Z. Wei, J. He, F. Yan, and L. Liu. "A two-step parameter optimization method for low-order model-based state of charge estimation." IEEE Transactions on Transportation
Electrification (2020). doi: 10.1109/TTE.2020.3032737.
Paper 2. X. Bian, L. Liu, J. Yan, Z. Zou, and R. Zhao. "An open circuit voltage-based model for state-of-health estimation of lithium-ion batteries: Model development and validation." Journal of Power Sources 448 (2020): 227401. doi: 10.1016/j.jpowsour.2019.227 401.
Paper 3. X. Bian, L. Liu, and J. Yan. "A model for state-of-health estimation of lithium ion batteries based on charging profiles." Energy 177 (2019): 57-65. doi: 10.1016/j.energy.2019.0 4.070.
Paper 4. X. Bian, Z. Wei, J. He, F. Yan, and L. Liu. "A novel model-based voltage construction method for robust state-of-health estimation of lithium-ion batteries." IEEE Transactions on
Industrial Electronics (2020). doi: 10.1109/TIE.2020.3044779.
Author’s contribution to the included publications
Xiaolei Bian made major contributions to these four included publications under the guidance of supervisors and with the collaboration of other co-authors. His contributions are mainly focus on methodology survey, data collection, code implementation, calculation, and draft writing.
Publications not included in this thesis
Paper 5. X. Bian, Z. Wei, W. Li, J. Pou, D. Sauer, and L. Liu. " State-of-health estimation of lithium-ion batteries by fusing an open-circuit-voltage model and incremental capacity analysis." IEEE Transactions on Power Electronics, submitted.
Paper 6. J. He, X. Bian, L. Liu, Z. Wei, and F. Yan. "Comparative study of curve determination methods for incremental capacity analysis and state of health estimation of lithium-ion battery." Journal of Energy Storage 29 (2020): 101400. doi: 10.1016/j.est.2020.101400. Paper 7. J. He, Z. Wei, X. Bian, and F. Yan. "State-of-health estimation of lithium-ion batteries using incremental capacity analysis based on voltage-capacity model." IEEE Transactions on
Acknowledgement
First and foremost, I would like to express my sincere gratitude to my main supervisor Prof. Longcheng Liu, for your continuous supports, patience, mentoring, and encouragements throughout my doctoral study. This thesis would not have been possible without your support and guidance. Your inspiring ideas and wisdom help me address many confusions in both of my study and daily life.
I would like to thank my co-supervisor Prof. Jinying Yan, for offering the valuable opportunity to study at KTH and also for all the supports and discussions. With your help, I have learnt a lot about energy storage in industry.
I am extremely grateful to Prof. Zhongbao Wei (from Beijing Institute of Technology, China) for all the valuable discussions and help. It was very enjoyable to collaborate with you in the past two years, and I have learned a lot of state-of-the-art knowledge about modeling and control of lithium-ion battery from you. I would also like to thank Dr. Jiangtao He and Prof. Fengjun Yan (from McMaster University, Canada) for all the cooperation and discussions.
I would like to thank China Electric Power Research Institute for doing the accelerated degradation tests and, in particular, to Dr. Chaoyong Hou for data collection and collation.
Thank you so much to Prof. Kerstin Forsberg and Prof. Göran Lindbergh, for your valuable comments about this thesis and the help of revision for the Swedish Abstract.
Thank you also to all my colleagues at the Department of Chemical Engineering for supports and creating a friendly working environment. A deep thank goes to Shuo Meng, Zhi Zou, Ruikai Zhao, Lai Zeng, Minyu Zuo, Xiaodong Li, Saiman Ding, Yang Zhang, Batoul Mahmoudzadeh, Raquel Rodriguez, Helen Winberg Wang, Matthäus Bäbler, Hyeyun Kim, and Maria Varini for your help, encouragement and all the happy time we have. Alexandra Rudyk Kinnander, Johanna Hagerman, and Nancy Hakala Malavé, thanks a lot for your kindest help and support for my student affairs.
I thank all my great friends for the company and support. Special thanks to our sporting team: Yan Xiang, Yucheng Deng, Haoran Ju, Changle Li, and Yuchuan Fan. With your company, I always feel fresh, energetic, and optimistic to keep going, and the days of cycling, ping-pong, and jogging are so amazing and unforgettable. Wenyuan Fan, Yangli Chen, Nan Zhao, Ge Li, Peng Yu, Lai Zeng, Liyun Yang, Xiaoqi Xu, Siying Wei, Di Fang, Yang Song, Boyu Xue, and Qiang Guo, thank you all for bringing so many wonderful moments in my life in Sweden, and the partying and gaming nights with you are always my sweetest memories.
Edward Michael Peters, Mahmood Alemrajabi, Sakarias Samak, Kivanc Korkmaz, and Jonas Ricknell, thank you all to make the Fika and beer moment so enjoyable and unforgettable. Special thanks to our lunch group: Tianhao Xu, Wujun Wang, and Simeng Tian, for all the beautiful lunch time we have at Brazilia Restaurang and AlbaNova restaurant. Special thanks to my lovely neighbors, Liang Zhang, Qiqiao Du, Jieyu Wu, and Yusen Liu for all the relaxing cooking and chatting time we have.
Yazhao Zhang, Peng Zhang, Tao Ren, Jing Gao, Youhe Zhang and Xiaoya Chen, my best friends in China, thank you so much for encouraging me all the time and help me go through many tough periods. I feel so lucky to have you as my friends.
Financial supports from the China Scholarship Council (CSC) and KTH Royal Institute of Technology are highly acknowledged.
I would like to express my deepest appreciation to my parents, brother, and sister, for your continual love, support, encouragement and help in my whole life.
Last but not least, I express my special appreciation and thanks to my beloved wife, Hongdi, for her unconditional supports, love, and continuous encouragement during the past seven years. I feel so blessed to have you in my life.
Contents
1 Introduction ... 1
1.1 Background ... 1
1.2 Purpose and contribution ... 5
1.3 Outline ... 6
2 A two-step parameter optimization method for SOC estimation ... 7
2.1 Experiment ... 7
2.2 Method development ... 9
2.3 Results and discussion ... 16
2.4 Summary ... 20
3 Two model-based methods for SOH estimation based on charging or discharging profiles ... 21
3.1 Experiment ... 21
3.2 Method development ... 23
3.3 Results and discussion ... 28
3.4 Summary ... 35
4 A fusion-type method for robust SOH estimation ... 37
4.1 Experiment ... 37
4.2 Method development ... 38
4.3 Results and discussion ... 40
4.4 Summary ... 51
5 Conclusions and outlook ... 53
5.1 Conclusions ... 53
5.2 Outlook ... 54
Nomenclature
Acronyms
1RC first-order RC model
2RC second-order RC model
3RC third-order RC model
BJDST Beijing dynamic stress test
BMS battery management system
CC constant-current
CV constant-voltage
DST dynamic stress test
ECM equivalent circuit model
EES electrical energy storage
EKF extended Kalman filter
EVs electric vehicles
FOIs feature-of-interests
FUDS federal urban driving schedule
GA genetic algorithm
HDS US06 highway driving schedule
IC incremental capacity
ICA incremental capacity analysis
KF Kalman filter
LFP LiFePO4
LIB lithium-ion battery
NMC LiNiMnCoO2
OCV open circuit voltage
PSO particle swarm optimization
RC resistor-capacitor
RMSE root-mean-square error
SOC state of charge
SOH state of health
Latin symbols
𝑎!,#$% constant
𝐴& area of the ith IC peak
𝐶 battery capacity
𝐶$'( estimated battery capacity
𝐶) constant
𝑓 state equation
𝐺 normally distributed Gaussian noise
ℎ output equation
𝐻*$+, peak height
𝐼! current
𝐼-./'$ current with artificial noises
𝑘 time index
𝑘0 adjusting coefficient
𝑘1 adjusting coefficient
𝑚 degree of the polynomial
𝑛 length of data
𝑛2 number of IC peaks
𝑂𝐶𝑉3& position of the ith IC peak
𝑄 charged/discharged capacity
𝐐! process noise covariance matrix
𝐐!∗ low boundary matrix of 𝐐
!
𝑄6+7 current maximum capacity
𝑄-.6 nominal capacity
𝑟 Pearson correlation coefficient
𝑅3 ohmic resistance
𝐑! measurement noise covariance matrix
𝐑!∗ low boundary matrix of 𝐑!
𝑅2 polarization resistance
𝑅𝑀𝑆𝐸89: RMSE of the state-of-charge estimation
𝑅𝑀𝑆𝐸; RMSE of the voltage estimation
𝑆 constant 𝑆*$+,' peak area 𝑆𝑂𝐶<)= state-of-charge estimation 𝑆𝑂𝐶><? state-of-charge measurement T temperature 𝐮! input vector 𝑉 terminal voltage
𝑉3 voltage over the ohmic resistance
𝑉3,! voltage over the ohmic resistance 𝑅3 at the time step 𝑘
𝑉@,! voltage over the networks of 𝑅@𝐶@ at the time step 𝑘
𝑉A,! voltage over the networks of 𝑅A𝐶A at the time step 𝑘
𝑉B,! voltage over the networks of 𝑅B𝐶B at the time step 𝑘
𝑉C,! voltage over the capacitor at the time step 𝑘
𝑉<)= estimated terminal voltage
𝑉! terminal voltage at the time step 𝑘
𝑉><? measured terminal voltage
𝑉-./'$ terminal voltage with artificial noises
𝑉DE open circuit voltage
𝑉2 polarization voltage
𝑉2,3 initial polarization voltage
𝑉*$+, peak voltage
𝑤! process noise
𝐱3 initial state vector
𝐱;34 initial estimate of state vector
𝐱! state vector
𝐱F lower bounds of initial state vector
𝐱G upper bounds of initial state vector
𝐲! output vector
∆t time interval
Greek symbols
𝜃 parameter set
𝜎 noise rate
𝜎H standard deviation of current measurement noises
𝜎; standard deviation of voltage measurement noises
𝜏 time constant
𝜑 tuning parameters
𝜑F lower bounds of the tuning parameters
𝜑G upper bounds of the tuning parameters
Chapter 1
Introduction
1.1
BackgroundElectrical energy storage (EES) technology has been developed rapidly in recent years, owning to its growing implementations in stationary energy storage systems and electric vehicles (EVs) [1, 2]. To alleviate the increasing concerns about climate change and energy crisis, the use of renewable energies, e.g., solar and wind powers is believed to be dramatically increased in the coming years. However, most of these renewable energy sources are highly intermittent. To integrate these renewable energies into the grid and to deliver stable and consistent power to the customers, reliable EES systems are highly required under such conditions [3]. Moreover, fossil fuels-based vehicles contribute about 14 % of carbon emissions globally [4]; therefore, profound
innovations are urgently required in the transportation to decrease the CO2 emission. EVs are
recognized as an important innovation in ground transportation which can significantly reduce the greenhouse gas emissions. The performance of EVs is highly depending on the reliability, safety, and functionality of the EES system. Noticeably, both renewable energies and EVs can contribute to the UN Global Goals for a sustainable society, especially for the achievement of Goal 7-Affordable and Clean Energy and Goal 13-Climate Action. Therefore, this thesis is believed to be closely related to the UN Sustainable Development Goals, by contributing to the development of renewable energies and EVs.
Lithium-ion batteries (LIBs) are the dominant EES solution in the automotive industry and are gaining popularity in the stationary energy storage system [5, 6]. This popularity originates from the outstanding advantages of LIBs, such as high energy density, high efficiency, long lifetime, low self-discharge rates, and no memory effect [7, 8]. However, LIBs contain both combustible material and oxidizing agent, and may catch fires or explode when the battery is suffered from abuse or misuse, such as overheating, over-charge, and short-circuit. The safety issue is still the major concern that hinders the market penetration of LIBs, witnessed by frequent battery-induced accidents happened around the world [9, 10]. Safe operation of LIBs requires a stable and effective battery management system (BMS). Among many tasks of BMS, the most important one is to monitor the key states of LIBs, such as the state of charge (SOC) and the state of health (SOH) [11, 12].
Accurate monitoring of the SOC and SOH is vital for the safe and efficient operation of LIBs. Serving like a fuel gauge in fossil fuels-based vehicles, SOC is a direct look at the remaining capacity of the battery and gives the user an indication of how long a battery will last before it needs recharging [13]. SOH acts, in the figurative sense, as a measure of the storage volume of the fuel tank, which gradually shrinks over time as the battery ages, and it gives the user an indication of how much energy a battery can still store or how much power a battery can now provide to perform its function [14, 15]. However, neither SOC nor SOH can be directly measured [16, 17]. Instead, they can only be inferred from models and diagnostic methods, based
on available measurements, e.g., voltage, current, and surface temperature [18]. In addition, these states are highly affected by the load dynamics and environmental conditions, which brings more difficulty for accurate estimation. Therefore, more accurate and robust methods are still required to estimate the SOC and SOH of LIBs.
1.1.1 SOC estimation
To estimate the SOC of LIBs, different methods have been proposed in the past decade. In general, these methods can be classified into five categories: Coulomb counting method, open circuit voltage (OCV)-based method, impedance spectroscopy-based method, data-based method, and model-based method [19].
Each of these methods has upsides and downsides. Coulomb counting is a simple and effective method for SOC estimation. This method is, however, vulnerable to the uncertainty of initial state, current measurement error, and battery self-discharge [20]. For the OCV-based method, the basic idea is to estimate the SOC through an OCV–SOC relationship with input of a measured OCV [21, 22]. There are two drawbacks with this method. First, to obtain the OCV– SOC relationship, extensive off-line tests are required. In addition, this relationship changes with ambient temperature and aging state which adds more uncertainties. Second, an accurate OCV is also hardly obtained in real applications, leading to more barriers for this method [23]. The impedance spectroscopy-based method can give more details about the inner mechanisms of LIBs. However, this method has the ex-situ nature, which challenges its potential for on-line applications [24]. The data-based method can be implemented through the techniques, e.g., artificial neural network, support vector machine, extreme learning machine, and fuzzy logic. This method can well map the relationship between the SOC and the available measurements, but it always requires intensive training and high computational effort [25, 26].
Compared with the above methods, the model-based method is the dominant one for on-line SOC estimation, owning to its high accuracy and reasonable computational cost [27]. As a most popular model-based method, the Kalman filter (KF)-based algorithms have been broadly utilized for state estimation in many areas. These algorithms are highly attractive for their enhanced robustness against the measurement and process noises [8]. In particular, among all the KF-based algorithms, the extended Kalman filter (EKF) is the most widely used one for SOC estimation. To employ the EKF to monitor the SOC, a battery model is required to describe the nonlinear behavior of the LIB [28-30].
The battery models for LIB can be generally categorized into two groups: electrochemical models and phenomenological models [31]. The electrochemical models can characterize the reactions inside the LIB based on the chemical kinetics and transport equations [32, 33]. However, the structures of these models are usually complex, and the involving parameters are difficult to identify. All these difficulties hinder their usage in on-line scenarios [34]. In contrast, the phenomenological models are more computationally efficient and hence more suitable for real applications. As one class of these models, the equivalent circuit models (ECMs) can well characterize the nonlinear performance of LIB with some electrical components, e.g., resistor, capacitor and voltage source [35]. It has been well demonstrated that the ECMs can describe the dominant dynamics of LIB with low computational cost. The ECMs are, therefore, widely used
for SOC estimation and especially in the on-line situations. The main limitation of ECMs is, however, that the model parameters are closely related to the aging status and the working conditions of the battery; the accuracy of the SOC estimation with ECMs may gradually deteriorate after certain cycles of operation [36].
Among the ECMs, the resistor-capacitor (RC) network-based models have outstanding balanced performance between complexity and accuracy [34]. In the RC network-based models, the first-order RC model (1RC), second-order RC model (2RC), third-order RC model (3RC), and the partnership for a new generation of vehicles model (PNGV) have excellent performance from the perspective of accuracy [8]. This conclusion was drawn from the comparison of different ECMs in reproducing the terminal voltage. However, the performance of different models in SOC estimation have not been well studied.
Despite have been widely explored, the model-based methods for SOC estimation still have many deficiencies waiting for further improvement. First, in most of the publications, the selection of battery models is very subjective without a critical comparison, let alone a general framework for performance optimization. In general, a complex model with higher order can get higher accuracy, but it usually costs more computational resources due to the high-order matrix operation. Second, the performance of SOC estimator is usually affected significantly by the involving parameters. For example, the covariance matrixes of EKF are critical for the accuracy of SOC estimation, and any unsuitable tuning of such parameters can cause large biases or even problem of divergence. However, it is still not clear how these parameters can be efficiently identified or optimized, excepting for the tedious trial-and-error method. Neither of these two problems has been well addressed in the available publications. To bridge these research gaps, Paper 1 proposes a unified, two-step algorithmic framework for improved SOC estimation of LIBs, whereby the problems of model selection and filter tuning of EKF can be well solved. 1.1.2 SOH estimation
It remains a challenging work to accurately estimate the SOH of LIBs, owning to the complex aging mechanisms [37, 38]. The aging mechanisms of LIBs can be roughly divided into three groups: loss of lithium inventory, loss of active material, and loss of conductivity [39, 40]; for more details, references [41-44] are recommended. In particular, aging modes of loss of lithium inventory and loss of active material, can lead to capacity fading; by comparison, power fading is mainly a result of loss of conductivity (the internal resistance is increased). The SOH can, therefore, be quantified according to the capacity decrease or the resistance increase, whereby we can reveal the energy or power capability of LIBs [39]. In this study, the stance adopted is to quantify battery degradation based on the capacity fading.
To estimate the SOH of LIB, many methods have been developed in the past few years. Some popular methods are the ampere-hour counting method, the machine learning-based method, the impedance spectroscopy method, the model-based method, and the incremental capacity analysis (ICA) method. Each of these methods has its advantages and disadvantages. The ampere-hour counting method is generally used for off-line calibration, because it requires the battery to be fully charged and discharged [45]. The machine learning-based method can well mimic the implicit and nonlinear relationship between the SOH and the measurable signals via extensive
data training. Nevertheless, it always demands a large amount of data for training and exhibits some intrinsic problems such as overfitting and poor generality [46, 47]. In contrast, the impedance spectroscopy measurement does not need data-learning or intensive computation, but remains mainly in ex-situ scenarios due to the need of strictly-defined spectral excitation [45]. By comparison, the model-based method and the ICA method are more promising for SOH estimation in real applications.
1.1.2.1 The model-based method
The model-based methods can be performed in both on-line and off-line applications, and usually have a good trade-off performance between accuracy and efficiency. These methods, therefore, are very popular in real applications [48]. Despite of these advantages, the model-based methods still have some drawbacks. First, such methods are highly sensitive to the modeling accuracy [29]. In general, a complex model with more components can get higher accuracy, but it usually cost more computational time and can bring the problem of over-fitting. Therefore, the balance between the accuracy and complexity is remaining a tricky work. Second, it is very difficult to guarantee the accuracy of SOH estimation under different conditions and over the lifetime of a battery [29].
For example, Guo et al. proposed a model-based method to estimate the SOH of LIBs [49]. This approach uses an ECM to characterize the constant-current (CC) portion of the charging profiles, from which a voltage transformation function with time-based parameters is derived. It was shown that, with only minor demand of learning from initial charging profile, an absolute
error of around 3 % was achieved for LiCoO2/LiNixMnyCozO2-based LIBs. This promising result
has encouraged us to apply the approach for SOH estimation of a LiFePO4 cell. It was, however,
failed because of the very different chemistries of LIBs and also the necessity of figuring out the information of OCV(SOC) function from the learning process. As a matter of fact, this key information is always contained in the transformation function even though it is apparently absent. Accordingly, we want to improve the mentioned method [49] to not only increase the accuracy, but also decrease the complexity.
With this understanding, in Papers 2 and 3, we developed two novel models for SOH estimation of LIBs. The proposed models can keep high accuracy under different conditions, especially with the learning-required strategy; therefore, the main deficiencies of model-based methods can be mitigated. The first model is an OCV-based model, which describes OCV simply as a function of SOC by a polynomial of high degree, with a lumped thermal model to account for the effect of temperature. It requires, however, a prior learning from the initial charging profile at the CC step. The second model is an ICA-based model, which applies the capacity model of [50] to directly define the dependence of SOC on OCV. It can be learning-unrequired, with all the parameters subject to certain constraints, to directly evaluate the actual capacity. 1.1.2.2 ICA method
The ICA method can well characterize the aging mechanisms of LIBs [51, 52]. In particular, using this method, the voltage plateaus on the capacity-voltage (Q-V) curve can be transformed to clearly identifiable incremental capacity (IC) peaks. The evolution of the area, location, and
amplitude of the IC peaks is closely related with the battery aging [53], and thus can serve as features of interests (FOIs) to estimate the SOH of LIBs [44, 54, 55]. Consequently, many attempts have been made to correlate the changes in FOIs to SOH estimation [56], by either directly mapping the FOIs to the SOH [57], or using data fusion methods, e.g., the sparse Bayesian model [58] and the Gaussian process regression method [14, 31]. For example, Zhang et al. [59] built a linear regression model between the battery capacity and FOIs to estimate the SOH of LIBs. Li et al. [37] proposed a new method to forecast battery health conditions, by relating the FOIs with SOH through Gaussian process regression.
Even with all of these efforts, it is still a challenging work to diagnose the health state of LIBs with the direct use of ICA [56]. First, the IC curves are highly sensitive to the measurement noise especially under the unexpected numerical condition of ∆𝑉 ≈ 0. Therefore, filtering of the Q-V curves is usually required; methods of e.g., interval sampling, moving average, improved center least squares method, and Gaussian filter are common choices under such conditions. However, the involved parameters must be tuned extensively. There is no mathematical guarantee for the fulfillment of curve smoothness and feature retention, which are both critical to the FOI extraction. To bridge these mentioned gaps, we propose a fusion-type method in Paper 4 for SOH estimation of LIBs [60]. It combines the model-based profile reconstruction and ICA-based state estimation, whereby the noise-induced unfavorable numerical conditions can be well alleviated.
1.2
Purpose and contributionThe purpose of this study is to contribute to the state estimation of LIBs. The accurate and robust monitoring of the key battery states is the foundation for battery management, and hence is vital for LIBs to work safely and efficiently. Even though widely explored, many research gaps are still existing. In this regard, the major contributions of this study are summarized as follows:
(1) This study proposes a unified, two-step algorithmic framework for SOC estimation of
LIBs, which has both a strong model compatibility and optimized performance. With this framework, the ECMs parameterizing and the covariance matrixes tuning can be achieved simultaneously. In particular, a novel boundary identification method is proposed to narrow the searching region, which can mitigate the premature problem and reduce the computation time.
(2) A novel OCV-based model is proposed to estimate the SOH of LIBs. It contains a lumped
thermal model to consider the effect of temperature. This model can accurately estimate the SOH under conditions with significantly fluctuated temperature. The effect of size and location of voltage window on the model’s accuracy is scrutinized, and we find that partial CC profile is enough to implement this model.
(3) Using an ECM to characterize the CC part of a charging/discharging profile, an
ICA-based model is proposed for SOH estimation of LIBs. This model can well consider the impact of non-linear and usage-dependent ageing effects, through implicitly propagating the OCV(SOC) information into a single-variable function V(Q) as battery ages. This model can apply either learning-required or -unrequired strategies to give accurate SOH estimation for different types of LIBs.
(4) A fusion-type SOH estimation method is developed, by combining the model-based profile reconstruction and the ICA-based state estimation. With the model-based voltage reconstruction method, we can mitigate the noise-induced unfavorable numerical conditions and reshape the disturbance-free IC curves. The effect of model order on the accuracy is scrutinized, offering deep insights to practical applications of the proposed method. The proposed method promises multi-fold benefits, including a high robustness to temperature change, noise corruption, cell inconsistency, and a satisfied generality to different battery chemistries.
1.3
OutlineThis thesis consists of five chapters. Chapter 1 gives the literature review and gap analysis, as shown above. The rest of the thesis is organized as follows:
Chapter 2 presents a novel method for on-line SOC estimation, which based on the work in Paper 1. The PSO algorithm is exploited to optimize the model parameters, and also be used to tune the error covariances of EKF. Within this PSO-based tuning framework, the searching boundary is derived by scrutinizing the error transition property of the test system. Further, the
proposed method is validated with experiments using LiNiMnCoO2 (NMC) cells.
Chapter 3 demonstrates two models to estimate the SOH of LIBs (based on the work in Papers 2 and 3). The first one is an OCV-based model, which describes the OCV as a function of the SOC by a polynomial, with a lumped thermal model to account for the effect of temperature. It requires a prior learning from the initial profile. The second one is an ICA-based model, which applies a capacity model to define the dependence of the SOC on the OCV as battery ages. It can be learning-free, with the parameters subject to certain constraints. Both models use a nonlinear least squares method for parameter identification. Moreover, experiments are performed to validate the proposed models.
Chapter 4 introduces a fusion-type SOH estimator by combining the model-based profile reconstruction and the ICA-based state estimation (based on the work in Paper 2 and Paper 4). A novel model-based voltage construction method is proposed to eliminate the unfavorable numerical condition and modify the IC curves. Leveraging the modified IC curves, a set of FOIs are extracted and evaluated, and several cautiously selected ones are used to estimate the SOH. The impact of model order on the estimation performance is scrutinized, to give insights into the parameterization in practical applications. Long-term cycling tests on different types of LIBs are used for validation and evaluation.
Chapter 2
A two-step parameter optimization method for SOC
estimation
In this chapter, a novel method is proposed for SOC estimation of LIBs (based on the work in Paper 1). First, the experiment is briefly introduced. Then, the method development is given in detail. Thereafter, the proposed method is implemented and validated in various scenarios.
2.1
Experiment
The Arbin BT2000 tester was used to conduct the experiment. In the test system, a thermal chamber was used to control the ambient temperature, and a computer was used to set load profiles and store data. The specifications of the tested cells are given in Table 2.1. The experiments included two kinds of tests, i.e., low-current test and dynamic test. More details about the experiments can be found in [23, 61].
Table 2.1. Specifications of the NMC cells.
Cell type 18650
Nominal capacity 2.0 Ah
Charge limit voltage 4.2 V
Discharge limit voltage 2.5 V
Maximum current 22 A (at 25 ℃)
Temperature range 0–50 ℃
In the low-current test, the cell was firstly charged in a constant current-constant voltage (CC-CV) manner: a CC (2 A) was applied to charge the cell before it got to the limit voltage (4.2 V), and then the same voltage was kept before the current is reduced to the limit value (0.02 A), at which the cell was fully charged with a SOC of 100 %. After left in an open-circuit state for 2 hours, the cell was discharged at a low CC (0.1 A) until it touches the lower limit voltage (2.5 V), at which the SOC is 0 %. Thereafter, the cell is rest for 2 hours, and then be fully charged with a low CC (0.1 A) to the upper limit voltage of 4.2 V (SOC = 100%). The profile of this test is illustrated in Fig. 2.1 (a). Based on this profile, the OCV-SOC relationship can be plotted [23], as shown in Fig. 2.1 (b).
In addition, four different dynamic tests were conducted, and they were dynamic stress test (DST), federal urban driving schedule (FUDS), US06 highway driving schedule (HDS), and Beijing dynamic stress test (BJDST). The profiles of these tests are shown in Fig. 2.2. Moreover, the DST was used to identify the parameters of ECMs, while the other tests were employed to validate the proposed method in SOC estimation.
Figure 2.1: (a) Profile of the low-current test, and (b) the SOC–OCV relationship (the average).
Figure 2.2: Profiles of four dynamic battery tests: (a) dynamic stress test, (b) federal urban driving schedule, (c) US06 highway driving schedule, and (d) Beijing dynamic stress test.
0 6 12 18 24 30 36 42 48 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 (a) Cur rent (A) Time (h) 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 Voltage (V) 0 20 40 60 80 100 2.4 2.8 3.2 3.6 4.0 4.4 (b) Voltage (V) SOC (%) Charge Discharge Average 0 2000 4000 6000 8000 10000 12000 -4 -2 0 2 4 6 (c) C urr ent ( A ) Time (s) 2.7 3.0 3.3 3.6 3.9 4.2 V ol tage ( V ) 0 2000 4000 6000 8000 10000 12000 -4 -2 0 2 4 6 (d) C urr ent ( A ) Time (s) 2.7 3.0 3.3 3.6 3.9 4.2 V ol tage ( V ) 0 2000 4000 6000 8000 10000 12000 -4 -2 0 2 4 6 (a) C urr ent ( A ) Time (s) 2.7 3.0 3.3 3.6 3.9 4.2 V ol tage ( V ) 0 2000 4000 6000 8000 10000 12000 -4 -2 0 2 4 6 (b) C urr ent ( A ) Time (s) 2.7 3.0 3.3 3.6 3.9 4.2 V ol tage ( V )
2.2
Method development
In this section, the development of the proposed method is introduced in detail. First, we show the structures of four battery models, and develop a PSO-based method to identify their parameters. Second, the fundamental principles and implementational process of EKF are illustrated and discussed. Finally, a novel method is proposed for the filter tuning of EKF. 2.2.1 Battery modeling
In this chapter, the ECMs of 1RC, 2RC, 3RC, and PNGV are utilized, and their schematic diagrams are shown in Fig. 2.3 [8, 34]. This figure shows that the RC network number could be one, two, or three, corresponding to the model of 1RC, 2RC, or 3RC, respectively. In addition, the PNGV model is a 1RC model coupled with one more capacitor between the ohmic resistance
𝑅3 and the 𝑅@𝐶@ network.
Figure 2.3: Schematic diagrams: (a) RC models, and (b) PNGV model. The current is defined as positive when the battery is in charging.
Each component in the structure of ECMs has their unique function in describing the battery behavior. The RC network is used to capture time-dependent effects of diffusion and charge-transfer processes [62], hence it can characterize the voltage’s exponentially decaying change [28]. By comparison, the ohmic resistance is used to track the instantaneous change of voltage. To increase the model accuracy, adding more RC networks in the model structure is a popular way [28], whereby the faster transients and dynamic behaviors of LIB can be well reflected [28, 63]. Nevertheless, a trade-off should always be considered between the accuracy and complexity, and the 1RC model is found sufficient for LIBs in most applications [63].
Based on the Kirchhoff law, the terminal voltages of the models 1RC, 2RC, 3RC, and PNGV are, (2.1) (2.2) (2.3) +
-+-
V0 R1 C1 + -V OCV(SOC, T)-
V1 + C3-
V3 +...
R3 I (a) R0 + -+-
V0 R1 C1 + -V OCV(SOC, T)-
V1 + I Cb-
Vb + (b) R0 0, 1, ( ) k oc K k k V =V SOC +V +V 0, 1, 2, ( ) k oc K k k k V =V SOC +V +V +V 0, 1, 2, 3, ( ) k oc K k k k k V =V SOC +V +V +V +V(2.4)
where 𝑉! is the terminal voltage; 𝑉DE is the open-circuit voltage; 𝑉3,! is the voltage over the
ohmic resistance 𝑅3; 𝑉@,!, 𝑉A,!, and 𝑉B,! are the voltages over the networks of 𝑅@𝐶@, 𝑅A𝐶A, and
𝑅B𝐶B, respectively; and 𝑉C,! is the voltage over the capacitor 𝐶C.
Moreover, the SOC in this study is defined as the ratio between the remaining capacity and the nominal capacity of a battery [64], i.e.,
(2.5)
where 𝑡 is the time, 𝑆𝑂𝐶3 is the initial SOC, 𝑄ID> is the nominal capacity, 𝜂 is the coulombic
efficiency, and 𝐼 is the current.
Some parameters in the Eqs. (2.1-2.4) are given as,
(2.6)
(2.7)
(2.8)
(2.9)
(2.10)
where 𝐼! is the current, and ∆t is the time interval.
After introducing the structures of these models, to use them in SOC estimation, we need identify their parameters firstly [65]. To alleviate the problem of premature, it is better to use heuristic optimization methods, e.g., the genetic algorithm (GA) and the PSO approach [66]. The PSO is initialized with some random candidates [67] and then uses an algorithm to search the global minimum for the defined fitness function. In addition, the PSO allows particles moving randomly in the problem space to search the global optimum. Compared with GA, PSO is more convenient to implement and has a more effective memory utilization[65]. The PSO is, therefore, used in this study to identify the model parameters.
The model parameters required to be identified for different models are summarized in Table 2.2. Moreover, minimization of the root-mean-square error (RMSE) of the estimated voltage is the target to evaluate the fitness of the identified parameters. Accordingly, the fitness function here is defined as,
0, 1, , ( ) k oc K k k b k V =V SOC +V +V +V 0 0 1 ( ) t ( ) nom SOC t SOC I t dt Q h = +
ò
0,k k 0 V =I R 1 1 1 1 1, 1, 1 1(1 ) t t R C R C k k k V V e I R e -D -D -= + -2 -2 2 2 2, 2, 1 2(1 ) t t R C R C k k k V V e I R e -D -D -= + -3 -3 3 3 3, 3, 1 3(1 ) t t R C R C k k k V V e I R e -D -D -= + -, , 1 1 b k b k k b V V I t C -= + D(2.11)
where denotes the RMSE of the voltage estimation; is the parameter set (shown in
Table 2.2); is the length of data; is the measured terminal voltage; and is the
estimated terminal voltage.
Table 2.2. Model parameters required to be identified.
Model name Parameters
1RC 𝑅!, 𝑅", 𝐶"
2RC 𝑅!, 𝑅", 𝐶", 𝑅#, 𝐶#
3RC 𝑅!, 𝑅", 𝐶", 𝑅#, 𝐶#, 𝑅$, 𝐶$
PNGV 𝑅!, 𝑅", 𝐶", 𝐶%
To make it easily to understand, the algorithm architecture to make parameter identification is shown in Fig 2.4 (a).
2.2.2 Fundamentals of EKF
EKF is a variant of the linear KF which can be used to solve nonlinear state estimation problems [68]. Reference [69] is recommended for more details about EKF. The state and output equations of EKF are given by,
(2.12)
(2.13)
(2.14)
(2.15)
where is the state vector, k is the time index, is the state equation, is the known input
vector, is the process noise with known covariance matrix , is the output vector, is
the output equation, and is the measurement noise with known covariance matrix .
To make it easily to follow, the procedure to implement the discrete-time EKF is summarized in Table 2.3. In addition, the state vectors of different models in the EKF are given in Table 2.4.
2 1 1 ( ) n ( ( ) ( , )) V k mea est RMSE V k V k n q =
å
= - q V RMSE q n Vmea Vest 1 1 1 ( , ) k= f k- k- +wk -x x u ( , ) k=h k k +vk y x u (0, ) k k w ! Q (0, ) k k v ! R k xf
uk k w Qk yk h k v RkTable 2.3. The main steps to implement the EKF. 1. Initialization:
,
2. For k=1,2,…, compute the partial derivative matrices: ,
3. Time update:
a. Time update of the state estimate: ,
b. Time update of the estimation-error covariance: 4. Measurement update:
a. Calculation of Kalman gain matrix: b. Calculation of output:
c. Measurement update of the state estimate:
d. Measurement update of the estimation-error covariance:
Notation: is the initial estimate of . The symbol of hat, i.e., “ ” refers the estimated values. The “ ” superscript refers the estimate is a posteriori. The “ ” superscript refers the
estimate is a priori. is the covariance of the estimation error of , and is the covariance
of the estimation error of .
Table 2.4. State vectors of different battery models. : : : : 0 0 ˆ+=E[ ] x x P0+=E[(x0-x xˆ0+)( 0-xˆ+0) ]T 1 1 ˆk k x f x + -¶ = ¶ F ˆ k k x h x -¶ = ¶ H 1 1 ˆk- f(ˆk+ , k ) - -= x x u ˆ k k x h x -¶ = ¶ H 1 1 1 1 T k k k k k - + - - - -= + P F P F Q 1 ( ) T T k= k- k k k- k+ k -K P H H P H R ˆ ( ,ˆ ) k k h k -= y x u ˆk ˆk k( k ˆk) += +- -x x K y y k ( - ) k k k += -P I K H P 0 ˆ+ x x0 ˆ + -k -P xˆk -k + P ˆk+ x 1RC xk=[V SOC1,k k]T 2RC xk=[V V SOC1,k 2,k k]T 3RC [ 1, 2, 3, ] T k =V V V SOCk k k k x PNGV [ 1, , ] T k= V V SOCk b k k x
Figure 2.4: Flowchart of the algorithms: (a) parameter identification, and (b) filter tuning of EKF. 2.2.3 Filter tuning
To implement the EKF, we need firstly conduct the filter tuning. Filter tuning is vital to guarantee the EKF functions well, but is always ignored in most publications [70]. The main purpose of
this process is to properly determine the initial estimation-error covariance matrix , the
process noise covariance matrix , and the measurement noise covariance matrix . The
detailed introduction about these three matrixes can be found in [19].
These three matrixes can theoretically be defined by considering the stochastic properties of the noises. However, the noise information is mostly unavailable, so these three matrices are normally determined by time-consuming trial-and-error method [71]. To mitigate this problem, a new PSO-based method for filter tuning is proposed. To decrease the number of identified
Initialize the 0-th generation swarm, and let i = 0.
Start
i ≤ 1000? No
Yes
Relative change in RMSESOC over last
60 iterations is less than 10-8 ? Tuning parameters φ Output Yes End No, i = i + 1 Input
PSO Main algorithm, with the fitness function: 𝑅𝑀𝑆𝐸SOC(𝜑) = , 1 𝑛/0𝑆𝑂𝐶mea(𝑘) − 𝑆𝑂𝐶est(𝑘, 𝜑 )< 2 𝑛 𝑘=1 OCV-SOC relationship
Table 4. Parameters identification of different battery models.
Model name 1RC 2RC 3RC PNGV 𝑅0 7.025 ∙ 10−2 6.962 ∙ 10−2 6.962 ∙ 10−2 7.021 ∙ 10−2 𝑅1 9.953 ∙ 10−3 9.487 ∙ 10−3 6.737 ∙ 10−3 9.849 ∙ 10−3 𝐶1 885.997 987.427 1390.441 880.088 𝑅2 − 1.115 ∙ 10−3 2.750 ∙ 10−3 − 𝐶2 − 655.938 3406.948 − 𝑅3 − − 1.115 ∙ 10−3 − 𝐶3 − − 655.474 − 𝐶b − − − 2.999 ∙ 10+6
The unite for resistances is Ohm, and for capacitances is Farad. Identified model parameters
FUDS Dataset
𝐑𝒌∗= 𝑘𝑅∙ 𝑑𝑖𝑎𝑔(𝜎𝑉2)
𝐐𝒌∗= 𝐿𝑘𝑄𝑘𝐿𝑇𝑘
Boundary determination Initialize the 0-th generation
swarm, and let i = 0. Start
i ≤ 1000? No
Yes
Relative change in RMSEV over last 60
iterations is less than 10-8 ? Identified parameters θ Output Yes End No, i = i + 1 Input
OCV-SOC relationship DST Dataset
PSO Main algorithm, with the fitness function: 𝑅𝑀𝑆𝐸V(𝜃) = , 1 𝑛/0𝑉mea(𝑘) − 𝑉est(𝑘, 𝜃)< 2 𝑛 𝑘=1 (a) (b) 0 + P k Q Rk
parameters, only the diagonal terms in and are considered, since the effect of off-diagonal terms is minor [70, 72].
The matrix can be calculated by,
(2.16)
Unfortunately, the value of is mostly unavailable in practice. The reasonable upper
and lower bounds of are, however, always available and hence and can be
approximated by,
(2.17)
(2.18)
Combine Eqs. (2.16-2.18), the matrix can be readily obtained.
For the matrixes and , the tuning aim is to minimize the RMSE of the SOC estimation.
Therefore, the fitness function of the PSO-based tuning method is,
(2.19)
where is the RMSE of the SOC estimation, is the tuning parameters, is the
measured SOC, and 𝑆𝑂𝐶<)= is the estimated SOC.
Afterwards, we need to define the bounds of these identified parameters; however, the bounds,
i.e., the lower bounds and the upper bounds are difficult to define properly. To solve
this problem, a novel method is proposed in this study. Based on Eq. (2.12), we know that when
the function is specified, the state vector is determined only by and . The sources
of the system noise are, therefore, from both the model structure and the input signal (the
current ). If we assume the models are perfect, then the source of system noise comes only
from the input signal. Therefore, based on the information of the test system, the low boundary
matrix can be obtained,
(2.20) (2.21) (2.22) k Q Rk 0 + P 0 ( [( 0 ˆ0)( 0 ˆ0) ]) T diag E += - + - + P x x x x 0 x ( )xu ( )xl x0 ˆ0 + x ( 0 ˆ0) + -x x 0 ˆ 0.5( u l) += + x x x 0 ˆ0 0.5( u l) + - = -x x x x 0 + P k Q Rk 2 1 1 ( ) n ( ( ) ( , ))
SOC k mea est
RMSE SOC k SOC k
n
j =
å
= - jSOC
RMSE j SOCmea
( )jl ( )
j
uf
xk xk-1 uk-1 k w uk k I wk * k Q * T k=L Q Lk k k Q ˆk k x f L w + ¶ = ¶ 2 ( ) k Q I Q = ×k diags
where is an adjusting coefficient (usually ); is the standard deviation of the current measurement noises.
Similarly, assume is the known standard deviation of voltage measurement noises, the
boundary matrix for can be calculated by,
(2.23)
where is an adjusting coefficient (usually ).
In practice, both the model error and the initialization error should all be considered. Hence,
the obtained matrices and could be used as the lower bounds in the PSO-based
filter tuning, and the upper bounds can be reasonably increased from . Moreover,
and can be utilized directly as and , if the accuracy is not highly demanding. To
make it easily to follow, the flowchart of the algorithm for filter tuning is shown in Fig. 2.4 (b). 2.2.4 Entire procedure summary
To make a summary, the schematic diagram of the proposed method is illustrated in Fig. 2.5. It should be noted that the OCV-SOC relation, parameterizing models, and tuning matrixes are determined in off-line and then input to the EKF algorithm. Afterward, the EKF can give on-line SOC estimation.
Figure 2.5: The entire procedure of the proposed method.
Q k ³1 sI V
s
* k R Rk * ( 2) k=k diagR× sV R R k ³1 * k Q * k R ( )jl ( )j
u jl Q*k * k R Qk RkExtended Kalman Filter
Time update Measurement update
State estimate 𝐱"𝑘+= 𝐱"𝑘−+ 𝐊𝑘(𝐲𝑘− 𝐲"𝑘) State estimate 𝐱"𝑘−= 𝑓𝑘−1(𝐱"𝑘−1+ , 𝐮𝑘−1) Estimation-error covariance 𝐏𝑘−= 𝐅𝑘−1𝐏𝑘−1+ 𝐅𝑘−1𝑇 + 𝐐𝑘−1 𝐲𝑘− 𝐲"𝑘 𝐲𝑘 + -𝐲"𝑘 𝐱"𝑘+ Estimation-error covariance 𝐏𝑘+= (𝐈 − 𝐊 𝑘𝐇𝑘)𝐏𝑘− 𝐏𝑘+ Kalman gain 𝐊𝑘= 𝐏𝑘−𝐇𝑘𝑇(𝐇𝑘𝐏𝑘−𝐇𝑘𝑇+ 𝐑𝑘)−1 Output 𝐲"𝑘= ℎ(𝐱"𝑘−, 𝐮𝑘) 𝐏𝑘− 𝐱"𝑘− 𝐊𝑘 𝐏𝑘−1+ 𝐮𝑘−1 𝐱"𝑘−1+ Initialization 𝐱"0+= 9𝑉1,0⋯ 𝑉3,0 𝑆𝑂𝐶0@𝑇 0 6 1218 243036 4248 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 Cur rent (A) Time (h) 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 Voltage (V) Low-Current test 0 20 40 60 80 100 2.4 2.8 3.2 3.6 4.0 4.4 Voltage (V) SOC (%) Charge Discharge Average OCV-SOC relation
Parameters of Battery models
Tuning results for EKF 0 2000 4000 6000 8000 10000 -4 -2 0 2 4 6 Curr ent ( A) Time (s) 2.7 3.0 3.3 3.6 3.9 4.2 Voltage ( V)
Dynamic Stress Test 0 2000 4000 6000 8000 10000
-4 -2 0 2 4 6 Curr ent ( A) Time (s) 2.7 3.0 3.3 3.6 3.9 4.2 Voltage ( V)
Federal Urban Driving Schedule Particle Swarm Optimization Particle Swarm Optimization
Model name 1RC 2RC 3RC PNGV 𝑅0 7.025 ∙ 10−2 6.962 ∙ 10−2 6.962 ∙ 10−2 7.021 ∙ 10−2 𝑅1 9.953 ∙ 10−3 9.487 ∙ 10−3 6.737 ∙ 10−3 9.849 ∙ 10−3 𝐶1 885.997 987.427 1390.441 880.088 𝑅2 − 1.115 ∙ 10−3 2.750 ∙ 10−3 − 𝐶2 − 655.938 3406.948 − 𝑅3 − − 1.115 ∙ 10−3 − 𝐶3 − − 655.474 − 𝐶b − − − 2.999 ∙ 10+6 1RC 𝐏0+ = 𝑑𝑖𝑎𝑔([2.5 ∙ 10−310−4]) 𝐐𝑘= 𝑑𝑖𝑎𝑔 ([2.5973 ∙ 10−310−14]) 𝐑𝑘= 10−2 2RC 𝐏0+= 𝑑𝑖𝑎𝑔 ([2.5 ∙ 10−32.5 ∙ 10−310−4]) 𝐐𝑘= 𝑑𝑖𝑎𝑔([2.0685 ∙ 10−310−210−14]) 𝐑𝑘= 10−2 3RC 𝐏0+= 𝑑𝑖𝑎𝑔([2.5 ∙ 10−32.5 ∙ 10−32.5 ∙ 10−310−4]) 𝐐𝑘= 𝑑𝑖𝑎𝑔 ([1.8999 ∙ 10−310−1410−210−14]) 𝐑𝑘= 10−2 PNGV 𝐏0+= 𝑑𝑖𝑎𝑔 ([2.5 ∙ 10−32.5 ∙ 10−1110−4]) 𝐐𝑘= 𝑑𝑖𝑎𝑔 ([1.6149 ∙ 10−5010−12]) 𝐑𝑘= 10−2
2.3
Results and discussion
In this section, the results are presented and discussed. First, we show the results of the proposed PSO-based method in parameter identification and filter tuning. Then, the proposed algorithmic framework is implemented with a dynamic test and the simulation results are illustrated. Finally, the robustness of the proposed method is validated under different scenarios.
2.3.1 Results of parameter identification
The PSO-based method can accurately identify the model parameters, and the identified parameters are summarized in Table 2.5. Using these identified parameters, the modeling results of different models are illustrated in Fig. 2.6. This figure that the RMSE is about 8.2 mV for each model; this low error validates the accuracy of the PSO-based method in parameter identification. For illustration purposes, the modeling result of the model 2RC is shown in Fig. 2.6 (b). It shows that the estimated voltage closely follow the measurement with error in the range of -0.03~0.03 V, which further validates the feasibility of the proposed method.
Table 2.5. Results of parameter identification for different battery models.
Parameter name 1RC 2RC 3RC PNGV 7.025∙10-2 6.962∙10-2 6.962∙10-2 7.021∙10-2 9.953∙10-3 9.487∙10-3 6.737∙10-3 9.849∙10-3 885.997 987.427 1390.441 880.088 1.115∙10-3 2.750∙10-3 655.938 3406.948 1.115∙10-3 655.474 2.999∙106 The unite for resistance is Ohm, and Farad for capacitance.
Figure 2.6: (a) The of the four models with the identified parameters, and (b) the
estimated and measured voltages of the 2RC model with the dataset of DST.
0 R 1 R 1 C 2 R - -2 C - -3 R - - -3 C - - -b C - - -8.1732 8.1659 8.1658 8.1556 1 RC 2 RC 3 RC PNGV 0 2 4 6 8 10 (a) RMSE V (mV) 0 2000 4000 6000 8000 2.7 3.0 3.3 3.6 3.9 4.2 Measured Estimated (b) Terminal voltage (V) Time (s) -0.03 0.00 0.03 0.06 0.09 0.12 Voltage err or (V) V RMSE
After identifying the model parameters, then we can conduct filter tuning for the EKF. The data collected at the initial stage of FUDS (0-1000 s) are used for filter tuning. The tuning results
of the matrixes , and for all the models can be found in [19].
To show the performance of the filter tuning, the SOC estimation after tuning is shown in Fig 2.7. It should be noted that the SOC estimation with PSO-based filter tuning has much higher accuracy, compared with the ones with the error tuning. This is because the trial-and-error method is very subjective and difficult to guarantee the accuracy. In contrast, this issue can be well alleviated by fully exploiting the early-stage data with the proposed method.
Figure 2.7: SOC estimation with different filter tuning methods: (a) the estimated results, and (b) the estimated errors
2.3.2 Results of SOC estimation
After finishing the parameter identification and filter tuning, the SOC of LIB can be estimated by EKF. Both the modeling result and the SOC estimation will be considered in the perspective of accuracy. When there is no initial SOC error, the simulation results are shown in Fig. 2.8. In this figure, all the estimated voltages are highly close with the measurements; the PSO-parameterized models can reproduce the battery dynamics with high accuracy. In regard of SOC estimation, the estimated results corresponding to the four models are highly similar with each other and well follow the measurements. In addition, for the SOC estimation error, the PNGV model has larger error compared with the other three models, as shown in Fig 2.8 (b).
To give a more detailed comparison, the error information is summarized in Table 2.6. The SOC estimations with models of 1RC, 2RC, and 3RC have extremely high accuracy
( ). By comparison, the estimation with PNGV model has a bigger error, but is
still confined to a low value of 0.14 %. These results highly validate the accuracy of the proposed method for SOC estimation.
0 + P Qk Rk 0 2000 4000 6000 8000 10000 10 20 30 40 50 60 70 80 90 Training area S O C ( % ) Time (s) Measured Estimated_PSO Estimated_trial-and-error (a) 0 2000 4000 6000 8000 10000 -3 -2 -1 0 1 2 3 E rr or ( % ) Time (s) PSO Trial-and-error (b) 0.01% RMSE<
Figure 2.8: Modelling results from EKF with the dataset of FUDS: (a) Terminal voltage estimation, and (b) SOC estimation.
Table 2.6. Errors of SOC estimation with different battery models.
Model name RMSE (%) Mean absolute error (%)
1RC 0.0075 0.0060
2RC 0.0075 0.0060
3RC 0.0076 0.0061
PNGV 0.1382 0.1302
When the initial SOC error is existed (SOC0 = 60 %), the proposed method is further
implemented, and the simulation results are shown in Fig. 2.9. We found it is interesting that the estimated voltages follow more closely with the measurements than those without initial SOC error. In other words, adding initial error can increase the accuracy of the models. This may be explained as that the EKF increases its trust on the model prediction against the measurement when the initialization error exists, leading to the improved modeling accuracy. Moreover, from Fig. 2.9 (b) and (c), we know that the estimated SOCs with different models can all converge to the measurement quickly, and then follow the measurement with good agreement.
To compare the accuracy, the errors of SOC estimation with different models are shown in Table 2.7. It is observed that all the four models can give high-accuracy estimation, with the RMSEs around 1.5 %. Therefore, by employing the proposed two-step PSO-based parameter optimization method, each one of these four models, albeit with different complexities, can be confidently used to estimate the SOC of LIB.
0 2000 4000 6000 8000 10000 2.7 3.0 3.3 3.6 3.9 4.2 4.5 (b) Termin al voltage (V) Time (s) Measured 1RC 3RC 2RC PNGV 10001250 1500 1750 2000 3.5 3.6 3.7 3.8 3.9 4.0 4000 4100 4200 4300 3.5 3.6 3.7 3.8 (1) (2) (2) (1) (a) 0 2000 4000 6000 8000 10000 -20 0 20 40 60 80 SOC (b) S OC ( % ) Time (s) Measured 1RC 2RC 3RC PNGV -0.4 0.0 0.4 0.8 1.2 1.6 E rr or ( % ) Error
Figure 2.9: Simulation results with the dataset of FUDS: (a) Voltage estimation, (b) SOC estimation, (c) SOC estimation at initial stage, and (d) SOC estimation errors.
Table 2.7. Errors of SOC estimation with different battery models when SOC0 = 60 %.
Model name RMSE (%) Mean absolute error (%)
1RC 1.459 1.157
2RC 1.458 1.157
3RC 1.447 1.147
PNGV 1.560 1.235
2.3.3 Robustness to different working conditions
Tests of HDS and BJDST are used to validate the robustness of the proposed method to uncertain dynamic conditions. Following the same procedure shown in Fig. 2.5, the proposed method is implemented with the HDS and BJDST datasets (no initial SOC error), and the results are shown in Tables 2.8 and 2.9, respectively.
These two tables indicate that all the models can give accurate estimation even under different dynamic operations. In addition, the SOC estimations based on the models of 1RC, 2RC, and 3RC have very similar and high accuracy. By comparison, the SOC estimation based on the PNGV model has relatively lower accuracy. In general, these results are consistent with those
0 2000 4000 6000 8000 10000 2.7 3.0 3.3 3.6 3.9 4.2 4.5 (b) Termin al voltage (V) Time (s) Measured 1RC 3RC 2RC PNGV 100012501500 17502000 3.5 3.6 3.7 3.8 3.9 4.0 4000 4100 4200 4300 3.5 3.6 3.7 3.8 (1) (2) (2) (1) (a) 0 2000 4000 6000 8000 10000 0 20 40 60 80 Initial SOC (a) SOC (%) Time (s) Measured 1RC 3RC 2RC PNGV 1000 1250 1500 1750 2000 66 68 70 72 74 4000 4100 4200 4300 50 51 52 53 (2) (1) (1) (2) (b) 0 20 40 60 80 100 60 65 70 75 80 85 Initial SOC (c) S OC ( % ) Time (s) Measured 1RC 2RC 3RC PNGV 0 2000 4000 6000 8000 10000 -5 0 5 10 15 20 E rr or ( % ) Time (s) 1RC 2RC 3RC PNGV (d)
obtained from the case using the dataset of FUDS, which further validates the robustness of the proposed method against uncertain dynamic conditions.
Table 2.8. Errors of SOC estimation with different models using the dataset of HDS.
Model name RMSE (%) Mean absolute error (%)
1RC 0.00689 0.00576
2RC 0.00682 0.00569
3RC 0.00682 0.00569
PNGV 0.18390 0.10589
Table 2.9. Errors of SOC estimation with different models using the dataset of BJDST.
Model name RMSE (%) Mean absolute error (%)
1RC 0.00446 0.00369
2RC 0.00445 0.00368
3RC 0.00445 0.00368
PNGV 0.19392 0.11402
2.4
Summary
A two-step PSO-optimized SOC estimator is developed, with the merits of high accuracy and low complexity. This method uses PSO to identify the model parameters and to tune the covariance matrices of EKF. To improve the performance of filter tuning, a novel method is proposed to define the boundaries of the covariance matrices. The proposed method is validated by experiments under different dynamic conditions.
Some main conclusions are summarized. First, by using the PSO-based parameter identification, all the models have high modeling accuracy, with RMSEs around 8.2 mV. Second, the PSO-based filter tuning can obviously improve the accuracy of SOC estimation, compared with the trial-and-error tuning method. Third, by using the proposed two-step PSO-optimization method, the SOC estimation based on different models are all highly accurate. Finally, the proposed method can fully exploit the potential of model-based SOC estimators; therefore, the use of the simplest 1RC model can still ensure the accuracy of SOC estimation, thus is recommended considering its lowest computational complexity.
